wpi pi computational fluid dynamics i
TRANSCRIPT
![Page 1: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/1.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics I
Reviewof
Fluid Dynamics
Instructor: Hong G. ImUniversity of Michigan
Fall 2001
![Page 2: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/2.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IOutline
Outline
Basic relations for continuum fluid mechanicsReynolds transport theoremDivergence theorem
Derivation of equations governing fluid flowConservation of massConservation of momentumConservation of energyConstitutive relations
![Page 3: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/3.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IBasic Relations
),( txφ
∫∫∫ ⋅+∂∂=
CSCVV
dSdVt
dVDtD
sys
)( nuφφφ
Reynolds Transport Theorem
Rate of changein system
For any vector or scalar function that represents a flow property
Rate of changein control volume
Flux through control surface
zw
yv
xu
ttDtD
∂∂+
∂∂+
∂∂+
∂∂=∇⋅+
∂∂= u
Material derivative
![Page 4: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/4.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics I
+ flux out
Basic Relations
At t=t CV
)(tφ
At t=t+dtCV
=CV
)( dtt +φ− flux in
![Page 5: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/5.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics I
The Divergence (Gauss) Theorem:Conversion of surface integral to volume integral
dVdSCVCS∫∫ ⋅∇=⋅ )()( φnφ
Basic Relations
![Page 6: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/6.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics I
Reynolds Transport Theorem:
Basic Relations
: System integral to control volume integral
∫∫∫ ⋅+∂∂=
CSCVV
dSdVt
dVDtD
sys
)( nuφφφ
∫
⋅∇+∂∂=
CV
dVt
)( uφφ
![Page 7: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/7.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
),(),( tt xx ρφ =
Conservation of Mass (Continuity)
If RTT yields
Since the equation holds for arbitrary control volume,
dVt
dVDtD
CVVsys∫∫
⋅∇+∂∂== )(0 uρρρ
0)( =⋅∇+∂∂ uρρt
![Page 8: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/8.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Or, in Lagrangian form,
Continuity Equation:
uuu ⋅∇+∇⋅+∂∂=⋅∇+
∂∂ ρρρρρ
tt)(
0=⋅∇+= uρρDtD
0or0)( =⋅∇+=⋅∇+∂∂ uu ρρρρ
DtD
t
![Page 9: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/9.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Conservation of MomentumIf RTT yields),(),( tt xuxφ ρ=
dVt
dVDtD
CVVsys∫∫
⋅∇+∂∂= )( uuuu ρρρ
∫∫ +⋅=CVCS
dVdS fnT ρ)(
[ ]∫ +⋅∇=CV
dVfT ρ
Surface force (stress) Body force (gravity)
Divergence theorem
![Page 10: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/10.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Conservation of Momentum
fTuuu ρρρ +⋅∇=⋅∇+∂∂ )(t
Alternatively, subtracting
⋅∇+∂∂⋅−⋅∇+
∂∂ )()( uuuuu ρρρρ
tt
)continuity(⋅u
fTuuu ρρρρρ +⋅∇==∇⋅+∂∂=
DtD
t)(
fTu ρρ +⋅∇=DtD
![Page 11: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/11.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Constitutive Relation – Stress Tensor
DUuT µµκ 2])32([ +⋅∇−+−= p
In tensor notation
Bulk viscosity = 0 (Stokes assumption)
∂∂
+∂∂+
∂∂−+−=
i
j
j
i
k
kijijij x
uxu
xupT µδµκδ )
32(
Unit tensor
001010100
[ ]T)()(21 uuD ∇+∇= (Deformation tensor)
![Page 12: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/12.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Conservation of Momentum (Final):
[ ] fDuu ρµµκρ +⋅∇+
⋅∇−∇+−∇= 2)
32(p
DtD
[ ]T)()(21 uuD ∇+∇=
![Page 13: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/13.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Conservation of Energy
From RTT, for
⋅+= uuxxφ
21),(),( tet ρ
dVeet
dVeDtD
CVVsys∫∫
⋅+⋅∇+
⋅+
∂∂=
⋅+ )
21()
21(
21 uuuuuuu ρρρ
dVdSdSCVCSCS∫∫∫ ⋅+⋅⋅+⋅−= fuTnunq ρ)()(
[ ]dVCV∫ ⋅+⋅∇⋅+⋅∇−= fuTuq ρ)(
Heat flux Work by stress
Body force work
Divergence theorem
![Page 14: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/14.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
fuTuquuuuu ⋅+⋅⋅∇+⋅−∇=⋅+⋅∇+
⋅+
∂∂ ρρρ )()
21()
21( ee
t
Total Energy Equation
0)( =⋅∇+∂∂ uρρt
fuTuquuuuu ⋅+⋅⋅∇+⋅−∇=⋅+∇⋅+
⋅+
∂∂ ρρρ )()
21(
21 ee
t
+⋅∇=∇⋅+
∂∂⋅ fTuuuu ρρρt
][ u:TT)(uT)(u ∇+⋅∇⋅=⋅⋅∇
)(: uTqu ∇+⋅−∇=∇⋅+∂∂ ete ρρ
![Page 15: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/15.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics IConservation Equations
Constitutive Relations
)(:)(: uUuT ∇−=∇ p
Tk∇−=q (Fourier’s Law)
),(),,( ρρ eTTepp == (Equation of State)
If viscous heating is neglected,
)()( uuu ⋅∇−=∇⋅+⋅−∇= ppp
)( uqu ⋅∇−⋅−∇=∇⋅+∂∂ pete ρρ
![Page 16: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/16.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics I
In Convective (Nonconservative) Form
Conservation Equations - Summary
0=⋅∇+ uρρDtD
fTu ρρ +⋅∇=DtD
)(: uTq ∇+⋅−∇=DtDeρ
![Page 17: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/17.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics I
In Conservative Form
Conservation Equations - Summary
0)( =⋅∇+∂∂ uρρt
fuuTu ρρρ +−⋅∇=∂∂ )(t
+⋅−⋅+⋅−∇=
⋅+
∂∂ qTuuuuuu )
21(
21 ee
tρρ
Discretized equations can satisfy the conservation properties more easily
![Page 18: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/18.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics I
In Integral Form (for fixed CV)
Conservation Equations - Summary
0=⋅+∂∂
∫∫CSCV
dSdVt
nuρρ
Useful in Finite Volume Methods
[ ]∫∫∫ ⋅−⋅+=∂∂
SCCVCV
dSdVdVt
)( nuunTfu ρρρ
∫
+⋅+⋅−⋅⋅+
SC
dSqe uuunTnu )21()( ρ
∫∫ ⋅=⋅+∂∂
CVCV
dVdVet
fuuu ρρ )21(
![Page 19: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/19.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics I
Compressible Inviscid Flows
Conservation Equations – Special Cases
0=∂∂+
∂∂+
∂∂+
∂∂
zyxtGFEU
=
Ewvu
ρρρρρ
U
+
+=
upEuwuv
puu
)(
2
ρρρρρ
E
+
+=
vpEvw
pvuvv
)(
2
ρρρρρ
F
++
=
wpEpvw
vwuww
)(
2
ρρρρρ
G
TchTceRTp pv === ,,ρ
ReTepRcv
)1(,)1(,1
−=−=−
= γργγor
![Page 20: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/20.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics I
Incompressible Flows
Conservation Equations – Special Cases
0=DtDρ
0=⋅∇ u
Continuity equation reduces to
0=⋅∇+ uρρDtD
(Divergence-free)
Momentum equation with constant viscosity, κ=0
fuuuu +∇+∇−=∇⋅+∂∂ 2ν
ρp
t
ρµν /= Kinematic viscosity
![Page 21: WPI PI Computational Fluid Dynamics I](https://reader030.vdocuments.mx/reader030/viewer/2022012813/61c4534870d67661a37625d3/html5/thumbnails/21.jpg)
PPPPIIIIWWWW Computational Fluid Dynamics I
Equation for Pressure
Conservation Equations – Special Cases
Taking divergence of the momentum equation
+∇+∇−=∇⋅+
∂∂⋅∇ fuuuu 2ν
ρp
t
Poisson’s equation
fuuuu ⋅∇+⋅∇∇+∇−=∇⋅⋅∇+⋅∇∂∂ )()()( 2
2
νρp
t
fuu ⋅∇+∇⋅⋅∇−=∇ ρρ )(2 p
which replaces the incompressibility condition.
(To be continued…)